Design a 3-bit Gray Code Counter Using D-flip-flops

Prerequisite : 3 bit down counter.

Overview:

• These types of counters fall under the category of synchronous controller counter.
• Here the mode control input is used to decide whether which sequence will be generated by the counter.
• In this case, mode control input is used to decide whether the counter will perform up counting or down counting.
• Designing of such a counter is the same as designing a synchronous counter but the extra combinational logic for mode control input is required.

Steps to design Synchronous 3 bit Up/Down Counter :

1. Decide the number and type of FF –

• Here we are performing 3 bit or mod-8Up or Down counting, so 3 Flip Flops are required, which can count up to  2³-1 = 7.
• Here T Flip Flop is used.

2. Write excitation table of Flip Flop –

Excitation table of T FF

3. Decision for Mode control input M –

• When M=0 ,then the counter will perform up counting.
• When M=1 ,then the counter will perform down counting.

4. Draw the state transition diagram and circuit excitation table –

State transition diagram for 3 bit up/down counting.

5. Circuit excitation table –

The circuit excitation table represents the present states of the counting sequence and the next states after the clock pulse is applied and input T of the flip-flops. By seeing the transition between the present state and the next state, we can find the input values of 3 Flip Flops using the Flip Flops excitation table. The table is designed according to the required counting sequence.

Circuit excitation table

If there is a change in the output state of a flip flop (i.e. 0 to 1 or 1 to 0), then the corresponding T value becomes 1 otherwise 0.

6. Find a simplified equation using k map –
Here we are finding the minimal Boolean expression for each Flip Flop input T using k map.

Simplified equation for K map

7. Create a circuit diagram –
The simplified expression for Flip Flops is used to design circuit diagrams. Here all the connections are made according to simplified expressions for Flip Flops.

3 bit synchronous up/down counter.

8. Timing Diagram –

Timing diagram for 3 bit synchronous Up/Down counter

Explanation :
Here -ve edge triggered clock pulse is used for toggling purpose.

Characteristics table of T FF

After every falling edge, when T = 1, the output state of Flip Flop will toggle.

• Initially  Q₃ = 0 , Q₂= 0 , Q₁= 0.

Case 1 : When M=0 ,then M'= 1

• T₃ = M'Q₂Q₁ + MQ'₂Q'₁ = Q₂Q₁.
• T₂ = M'Q₁ + MQ'₁= 1.Q₁= Q₁.
• T₁= 1.
• Because T₁= 1, therefore FF1 output state toggles for every falling edge.
  The output state of FF 2 will toggle when Q₁ = 1  and the falling edge of the clock pulse occurs.
  The output state of FF 3 will toggle only when Q₂.Q₁= 1 and the falling edge of the clock pulse occurs.
• In this way, after every falling edge, state transition takes place and we can get our desired counting sequence.

Case 2 : When M=1 ,then M' =0

• T₃ = M'Q₂Q₁+MQ'₂Q'₁ = Q'₂Q'₁
• T₂ = M'Q₁+ MQ'₁= 1.Q₁= Q'₁.
• T₁= 1.
• Because T₁= 1,therefore FF1 output state toggles for every falling edge.
  The output state of FF 2 will toggle when Q'₁ = 1  and the falling edge of the clock pulse occurs.
  The output state of FF 3 will toggle only when Q'₂.Q'₁= 1 and the falling edge of the clock pulse occurs.
• In this way, after every falling edge, state transition takes place and we can get our desired counting sequence.

Design a 3-bit Gray Code Counter Using D-flip-flops

Source: https://www.geeksforgeeks.org/synchronous-3-bit-up-down-counter/

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